Vortex Dipole in a BEC with dipole-dipole interaction

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by  Cem Yuce

Vortex Dipole in a BEC with dipole-dipole interaction
C. Yuce, Z. Oztas
Physics Department, Anadolu University, Eskisehir, Turkey∗
(Dated: January 4, 2011)
WeconsidersingleandmultiplychargedquantizedvortexdipoleinanoblatedipolarBoseEinstein
condensateintheThomas-Fermi(TF)regime. Wecalculatethecriticalvelocityfortheformationof
apairof vorticesofoppositecharge. Wefindthatdipolarinteractions decreasethecritical velocity
for a vortex dipole nucleation.
1
1
PACSnumbers: 03.75.Nt,03.75.Kk,67.85.De
0
2

n I. INTRODUCTION
a
J
Avortexdipoleisapairofvorticesofequalandoppositecirculationsituatedsymmetricallyabouttheorigin. Under
3
linear motionof a localizedrepulsiveGaussianpotential, a vortexpair formationwith opposite circulationis possible

] if the potential is moved at a velocity above a critical value [1]. In the experiments [2, 3], a laser beam focused
s on the center of the cloud was scanned back and forth along the axial dimension of the cigar shaped condensate.
a
Vortices were notobserveddirectly, but the strongheating only abovea criticalvelocity wasmeasured. Itwas shown
g
- that the measurement of significantly enhanced heating is due to energy transfer via vortex formation [4]. Recently,
nt experimental observations of singly and multiply chargedquantized vortex dipoles in a highly oblate BEC with 87Rb
a were reported by Neely et al. [5]. In the experiment, vortex dipoles were created by forcing superfluid around a
u repulsive Gaussian obstacle using a focused blue-detuned laser beam. The beam was initially located on the left of
q thetrapcenterandtheharmonicpotentialwastranslatedataconstantvelocityuntiltheobstacleendsupontheright
.
t ofthetrapcenter. Atthesametime,theheightoftheobstacleislinearlyrampedtozero,leadingtothegenerationof
a
avortexdipolethatisunaffectedbythepresenceofanobstacleorbyheatingduetomovingtheobstaclethroughthe
m
edges of the BEC where the local speed of sound is small. Vortex dipoles were observed to survive for many seconds
-
in the condensate without self-annihilation. The experiment also provided evidence for the formation of multiply
d
n charged vortex dipoles. The authors in [5] noted that the theoretical predictions of critical velocity for vortex pair
o formation in [6] are in good agreement with the experimental results. The critical velocity is given by the minimum
c of the ratio of the energy to the momentum of the vortex dipole
[

E(I)
1 v =min( ) (1)
c
v I
1
4 where E(I) is the energy of an elementary excitation with linear momentum (or impulse) I [7]. When the object
5 moves at a velocity above a critical value, the superfluid flow becomes unstable against the formation of quantized
0 vortices, which give rise to a new dissipative regime [8–10]. Pairs of vortices with opposite circulation are generated
1. at opposite sides of the object. Recently, instead of removing the trapping potential and expanding the condensate
0 to makethe vortexcoresopticallyresolvable,Freilichetal. experimentallyobservedthe real-timedynamics ofvortex
1 dipoles by repeatedly imaging the vortex cores [11]. Vortex tripoles have also been observed experimentally [12].
1 Several theoretical investigations have been reported for the generation [13–17], stability [18, 19], and stationary
:
v configurationsof vortex dipoles [20, 21]. In addition, fully analytic expressions of the angular momentum and energy
i of a vortex dipole in a trapped two dimensional BEC were obtained [22].
X 52
ThesuccessfulrealizationofBose-Einsteincondensationof Cr atomshasstimulatedagrowinginterestinthe study
r of BEC with nonlocal dipole-dipole interactions [23–29]. A particular interest is the vortex structures in dipolar
a
condensates [30–46]. In contrastto the isotropic characterof contact interaction,long rangedand anisotropicdipole-
dipole interaction has remarkable consequences for the physics of rotating dipolar gases in TF limit. It was shown
that,inaxiallysymmetrictrapswiththeaxisalongthedipoleorientation,thecriticalangularvelocity,abovewhicha
vortexis energeticallyfavorable,is decreaseddue to the dipolar interactionin oblate traps[42]. It wasdiscussedthat
the effect of the dipole-dipole interaction is the lowered precession velocity of an off-center straight vortex line in an
oblatetrap[43]. Ouraiminthepresentworkistocalculatethecriticalvelocityforvortexdipoleformationinadipolar
oblate BEC in TF regime. This paper is structured as follows. Section II reviews Bose-Einstein condensates with
dipole-dipole interaction. Section III investigates the critical velocity in the presence of the dipole-dipole interaction.
The last section discusses the results.
2
II. DIPOLAR BEC
Consider a BEC of N particles with magnetic dipole moment oriented in the z direction. In the mean field theory,
the order parameter ψ(r) of the condensate is the solution of the Gross-Pitaevskiiequation (GPE) [28, 29]
2
¯h
2+V +g ψ(r)2+Φ ψ(r)=µψ(r) (2)
T dd
−2m∇ | |
(cid:18) (cid:19)
2
4π¯h a
s
where µ is the chemical potential, g = , a is the s-wave scattering length, V is the trap potential
s T
m
1
2 2 2 2
V = mω ρ +γ z (3)
T
2 ⊥
(cid:0) (cid:1)
γ is the trap aspect ratio, and Φ (r) is the dipolar interaction
dd
3 1 3cos2θ
Φdd(r)= 4πgεdd d3r′ −r r 3 |ψ′(r)|2 (4)
Z | − ′|
where r r′ is the distance between the dipoles, θ is the angle between the direction of the dipole moment and the
−
vector connecting the particles, and the dimensionless quantity ε is the relative strength of the dipolar and s-wave
dd
interactions [47]. The BEC is stable as long as 0.5 < ε < 1 in the TF limit [30, 47]. In the regime ε < 0, the
dd dd
−
dipolar interaction is reversed by rapid rotation of the field aligning the dipoles [31, 32].
We shall use scaled harmonic oscillator units (h.o.u.) for simplicity. In this system, the units of length, time, and
¯h 1
energy are , and h¯ω , respectively. Hence the GP equation in h.o.u reads
2mω 2ω ⊥
r
⊥ ⊥
2+V +g ψ (r)2+Φ ψ (r)=µψ (r) (5)
−∇ T′ ′| ′ | ′dd ′ ′ ′
(cid:0) (cid:1)
where the dimensionless interaction parameter g is given by
′
2mN 2mω
g′ = ¯h2 ¯h ⊥ g (6)
r
and Φ′dd =Φdd(g→g′). The normalization of ψ′(r) chosen here is d3r|ψ′(r)|2 =1.
Z
The equation (5) is an integro-differential equation since it has both integrals and derivatives of an unknown wave
function. The solution of this equation was presented by Eberlein et al. in TF regime [47]. They showed that a
parabolic density remains an exact solution for an harmonically trapped vortex-free dipolar condensate in TF limit:
ρ2 z2
n(r)=n0 1− R2 − L2 (7)
(cid:18) (cid:19)
µ
′
where n0 = and R and L are the radial and axial sizes of the condensate, respectively. The condensate aspect
g
′
R
ratio κ is defined as κ= . In the absence of dipolar interaction, the condensate aspect ratio κ and the trap aspect
L
ratioγ match. κ decreaseswith increasingε in anoblate trap. Hence, for ε >0 (ε <0), κ<γ (κ>γ)[30, 31].
dd dd dd
In h.o.u, the dipolar mean-field potential inside the condensate in the TF approximation is given by [47]
2 2 2 2
ρ 2z 3ρ 2z
Φdd(r)=n0g′εdd R2 − L2 −f(κ) 1− 2R2− L2 (8)
(cid:18) (cid:18) − (cid:19)(cid:19)
where f(κ) for oblate case, (κ>1), is given by
2+κ2(4 6 arctan√κ2 1 )
f(κ)= 2−(1 κ2√)κ2−1− (9)
−
3
III. VORTEX DIPOLE
VorticescanbenucleatedinaBECbyalocalizedpotentialmovingatavelocityaboveacriticalvalue[48]. Vortices
with opposite circulationare generatedatopposite sides ofthe condensate. The motionof eachvortexarisesfromits
neighbor. In the presence of weak dissipation, the two vortices slowly drift together and annihilate when the vortex
separation is comparable with the vortex core size.
The critical velocity expression for a vortex dipole formation derived by Crescimanno et al. [6] was found in good
agrement with the experimental observation [5]. We now extend their approach to include dipolar interaction. Con-
sidera pairofsinglevorticeswithoppositecharge. We supposethe vorticesarelocatedsymmetricallyaboutthe trap
center. Let the distance between the cores be represented by d. Substituting ψ(r)= n(r)eiφ into the GP equation
and equating imaginary and real terms leads to the following hydrodynamics equations:
p
ρ2+γ2z2
2+( φ)2+ +g n+Φ √n=µ√n (10)
′ dd ′
−∇ ∇ 4
(cid:18) (cid:19)
√n 2φ 2 φ √n=0 (11)
− ∇ − ∇ ·∇
The term 2√n in the equation (10) is negligible within TF approximation. The ansatz for the phase function for a
∇
vortex dipole is [6]
sinθ d sinθ+ d
φ(r)=l arctan( − 2ρ) arctan( 2ρ ) (12)
cosθ − cosθ !
¯h
where ρ and θ are the polar coordinates, and l is vorticity. The condensate velocity is given by v(r)= φ(r).
m∇
We assume that the vortices are far enough from each other, but near enough to the trap center. The ansatz is
1
2
equivalent to requiring <d <µ′ [6]. Using (12) we find
µ
′
1
l2d2 2
φ = (13)
|∇ | ρ2d2cos2θ+(ρ2− d42)2+η!
l2d2
where η = . Excluding the vortex core regions from the domain complicates the analytic evaluation of energy
µ
′
and impulse precisely where the TF approximation fails. To prevent this difficulty, η is added to the denominator
of (13) [6]. This regulated expression is confirmed by the observation that for vortex pair not too far from the trap
center (d2 <µ), the contribution to √n(r) from the kinetic energy term φ2 is never larger than µ [6].
′ ′
|∇ |
It is reasonable to approximate the TF density in h.o.u. by
1 ρ2+κ2z2
n(r)= µ (14)
′
g − 4
′ (cid:18) (cid:19)
2 2 4 2
The correction due to the term φ is at the order of d /R , which is very small compared to R in h.o.u. Let us
|∇ |
calculate the total energy and the total impulse of the condensate. The expression of total energy of the condensate
in h.o.u is given by
(ρ2+γ2z2) g Φ
E = d3r ( √n)2+(√n φ)2+ n+ ′n2+ ′ddn (15)
∇ ∇ 4 2 2
Z (cid:18) (cid:19)
i¯h
The total expression of impulse using the momentum of the condensate P= [( ψ )ψ ψ ψ] is
∗ ∗
2 ∇ − ∇
I = d3rP =h¯ d3r n(r) φ (16)
| | |∇ |
Z Z
The critical velocity for vortex pair creation using the Landau criterion is defined as [6]
El E0
v = − (17)
c
I
l
4
2.2
Εdd=0
2.0 Εdd¹0
1.8
vc
1.6
1.4
0 1 2 3 4 5 6
d
FIG.1: Thecritical velocity,vc (mm/s),forεdd =0(solid)andεdd =0.15(dashed)asafunction ofdinunitsofh.o.u. foran
oblate trap with γ =5
l=1
3.5 l=2
3.0
vc
2.5
2.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Ε
dd
FIG. 2: The critical velocity, vc (mm/s), for d=1 in units of h.o.u. as a function of εdd for an oblate trap with γ =5. The
solid (dashed) curveis for singly (doubly)quantized vortex dipole.
Here El and Il are the energy and impulse of vortex state whereas E0 is the energy of non-vortex state. Analytical
evaluation of these energy, impulse and critical velocity functions were performed by Crescimanno et al. in two
dimensionsforanondipolarcondensate[6]. Inthisstudy,wewillcalculatethecriticalvelocityofadipolarcondensate
in an oblate trap. As the resulting equations for v are complicated, it is necessary to obtain them numerically for
c
given a , ε , κ. We compare the critical velocities of dipolar and non-dipolar condensates.
s dd
IV. RESULTS
In this paper, within the TF regime we perform a numerical calculation of critical velocity for a vortex pair
52
formation in a dipolar BEC. We examine dipolar gas containing 150000 Cr atoms in an oblate trap with trap
frequencies ω =2π 200 rad/s, ω =2π 1000 rad/s, so the trap aspect ratio is γ = 5. The magnitude of the
z
scattering len⊥gth for×52Cr is 105a (a is×the Bohr magneton). We assume that vortex separation d satisfies the
B B
1
2
condition <d <µ′. The chemical potential is approximately 42 in h.o.u. for εdd =0 and changes very slightly
µ
′
with d. The dipole-dipole interaction decreases the chemical potential. Below, we perform numerical integration of
1.0
l=1
0.8 l=2
0.6
vc c
0.4
0.2
0.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Ε
dd
vc
FIG. 3: The ratio of critical velocity to the speed of sound, , for d=1 in units of h.o.u. as a function of εdd for an oblate
c
trap with γ =5. The solid (dashed) curveis for singly (doubly) quantized vortex dipole.
5
energy and impulse functional (15,16) to find the critical velocity of vortex pair formation (17).
Itiswellknownthatinanoblatetrap,condensateaspectratio,κ,decreaseswithincreasingε andthedipole-dipole
dd
interactionenergyispositive. Minimizingtheenergyfunctionalofthedipolarcondensate,wefindthatthecondensate
aspectratioincreasesslightlywithdandtheradiusRisalmostthesameforalld. Forexample,κisbetween4.73and
4.76 when d is between 0.5 and 6, respectively for ε =0.15. The kinetic energy changes significantly with vortices
dd
separation d, while the other terms in the energy expression change slightly with d. The impulse of vortex state, I,
2
depends strongly on d and ε since it scales as d/R (13,16). It decreases with ε , since the dipolar interaction
dd dd
stretches the cloud radially in an oblate trap.
The critical velocity v decreases with increasing separation d for a non-dipolar condensate [6]. This is because the
c
TF density is a maximum at the trap center and reduces with the distance away from the trap center. We expect
that the criticalvelocity decreaseswith d alsofor a dipolar BEC since parabolicform ofdensity retains inthe case of
dipolar interaction. Fig-1 plots the critical velocity v as a function of the distance between the vortices for ε =0
c dd
(solid curve) and for ε = 0.15 (dashed curve). The critical velocity is between 1.25 2.15 mm/s for ε = 0.15,
dd dd
−
and 1.27 2.24 mm/s for ε = 0 in the range 6>d>0.5. The critical velocity difference between dipolar and
dd
−
non-dipolar condensates becomes smaller as d is increased. This is because κ increases with d. As can also be seen
fromthe figure,the inclusionofdipolarinteractiondecreasesthe criticalvelocityfora fixedvalueofd. Thisis always
true for positive values of ε . In the case of negative values of ε , the effect of dipole-dipole interaction is the
dd dd
increasedcriticalvelocity. Fig-2 shows the critical velocity as a function of ε for fixed d=1. The solidcurve shows
dd
singly quantized vortex dipole while the dashed curve shows doubly quantized vortex dipole. The effect that v is
c
decreasedwithincreasingε inanoblatetrapisthefirstmainresultofthispaper. Itisenergeticallylessexpensiveto
dd
nucleate a vortex in an oblate dipolar Bose-Einstein condensate than in a condensate with only contact interactions.
At firstsight, this mightseemcounterintuitive since the dipole-dipole interactionenergyis positive inan oblate trap.
It is remarkable to note that although dipolar interaction is positive for an oblate trap, the excess dipolar energy is
negative.
The nucleation of multiply charged vortex dipoles was observed for trap translation velocities well above v in the
c
experiment[5]. Furthermore,itwasobservedthatthevorticesexhibitperiodicorbitalmotionandvortexdipolesmay
exhibit lifetimes of many seconds, much longer than a single orbital period [5]. Fig-2 compares the critical velocity
for singly and doubly quantized vortex dipole. As expected, v is bigger for doubly quantized vortices.
c
On the investigation of a vortex dipole, not only v , but also the ratio v /c is of importance. Here c is the speed of
c c
sound [27]
c= n0g(1+ǫ (3cos2α 1)) (18)
dd
m −
r
where α is the angle between directions of the wave vector and the dipoles. Suppose the direction of the phonon
wave vector is perpendicular to the orientation of the dipoles (α=π/2). In this case, the speed of sound becomes
n0
c= g(1 ǫ ). Remarkably,both v and c decreasewith ε . However,the ratio v /c increaseswith ε . As ε
dd c dd c dd dd
m −
r
goes to one, the critical velocity approaches to the speed of sound. In Fig-3, we plot v /c versus ε for singly (solid
c dd
curve) and doubly quantized vortex dipoles (dashed curves) for fixed d = 1. The effect that v /c is increased with
c
increasing ε in an oblate trap is the second main result of this paper. The ratio v /c for singly quantized vortices
dd c
arefoundtobe between0.16 0.31fordipolarcondensatewithε =0.15and0.15 0.28fornon-dipolarcondensate
dd
− −
in the range 6>d>0.5. As expected, the ratio v /c increases for doubly quantized vortices.
c
In this paper, we have studied single and multiply quantized vortex dipoles in an oblate dipolar Bose Einstein
condensate. We have shown that v is decreased while v /c is increased with increasing ε in an oblate trap. The
c c dd
dynamics of vortex dipole in a dipolar BEC is worth studying. Helpful discussions with A. Kilic are gratefully
acknowledged.
∗
Electronic address: [email protected]
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